I need to find the conditions on $A$ and $B$ so that for every bounded sequence $b_t$ (except the zero sequence) the following expression will be negative:
$\sum\limits_{t=0}^\infty \beta^t b_t^2 A+2\sum\limits_{i=0}^\infty \sum\limits_{j<i} \beta^i\delta^{i-j-1} b_i b_j B$
[$\beta,\delta \in (0,1)$ and constant. $A,B$ can depend on them.]
I know that in general, this is equivalent to checking that the appropriate operator is negative definite, but I'm not sure how to check it in this case in a way that will produce the required conditions.
Any suggestions and ideas would be appreciated.
(This is the second derivative of some optimization problem).